Combinatorics - The Students and Grades Problem

Question

Suppose there are $20$ students in a classroom, $10$ males and 10 females. How many ways can a teacher distribute exactly $5$ As, $6$ Bs, $5$ Cs, $2$ Ds, and $2$ Fs to the students?

My Logic: $(20 C 5)(15 C 6)(9 C 5)(4 C 2)(2 C 2)$ seems the like the most intuitive answer but then this answer ignores the number of males and females in the class. Rather, if we treat the whole classroom as our sample, then we avoid any extra casework. I may be missing something here though so feel free to enlighten me.

Answer 1

Your result looks good.

Another way to obtain the same result is as follows:

• Any arrangement of the $20$ students can be interpreted as a possible assignment of grades: $20!$
• Now, cancel out the permutations among each group of students who get the same grade: $5!\cdot 6!\cdot 5!\cdot 2!\cdot 2!$

All together: $$\frac{20!}{5!\cdot 6!\cdot 5!\cdot 2!\cdot 2!}$$